Inverse Problems for Sturm–Liouville-Type Differential Equation with a Constant Delay Under Dirichlet/Polynomial Boundary Conditions

The topic of this paper are non-self-adjoint second order differential operators with constant delay generated by $-y +q(x)y(x-\tau)$ where potential $q$ is complex-valued function, $q\in L^{2}[0,\pi]$. We study inverse problems of these operators for $\tau\in\left[\frac{2\pi}{5},\pi\right)$. We investigate the inverse spectral problems of recovering operators from their two spectra, firstly under Dirichlet-Dirichlet and secondly under Dirichlet-Polynomial boundary conditions. We will prove theorem of uniqueness, and we will give procedure for constructing potential. In the first case for $\tau\in\left[\frac{\pi}{2},\pi\right):$ we will show that Fourrier coefficients of a potential are uniquely determined by spectra. In the second case for $\tau\in\left[\frac{2\pi}{5},\frac{\pi}{2}\right):$ we will construct integral equation under potential and we will prove that this integral equation has a unique solution. Also, we will show that other parameters are uniquely determined by spectra.